Black holes and space-time

Black holes can pull photons in although they move at the speed of light. So, does this means that black holes pull space-time in faster than light and if so, why can space-time "travel" faster than light?

Welcome to PF;
Space-time does not "travel" (not the way you seem to be talking about anyway[*]). The BH is a region of extremely curved space-time so that, when viewed from a distance, objects tend to get sucked in (they "roll down" the curve).

It is a little like a wagon rolling on the ground - a steep-sided hole can "suck down" even the fastest wagon, but the ground does not have to move to do this.

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[*] there are situations where two objects can have their separation change faster than the speed of light ... the expansion of the Universe is one example.
You have to be careful not to let this confuse you. At this stage you just need to wrap you mind around the concept of space-time.

Black holes can pull photons in although they move at the speed of light. So, does this means that black holes pull space-time in faster than light and if so, why can space-time "travel" faster than light?

As Simon Bridge has pointed out, this is not space-time traveling.

Interestingly however, space can and does 'expand' faster than the speed of light. This is also not a 'travel', and is not prevented by relativity theory.

In relativity, all speed has to be relative to something. When we talk of the "speed of light", we mean speed relative to a local particle with mass. Light trying to escape from a black hole is travelling at exactly the speed of light relative to a local observer who can't escape either and who is falling in faster than the light is.

<sigh> I know - but Dr Greg responded to my comment, saying I got it wrong, using outward directed example to illustrate. But I was talking about inwards directed light. We spend all this time telling students that velocity is a vector right? [mumble: This was supposed to be a one-comment aside mutter grumble] :( Please lets continue in private or we'll hijack the thread.

I'm sorry if you got that impression, but I never actually said that. I was responding to the original questioner, and my post wasn't even consecutive to yours, and made no mention of it. As PAllen said, I was talking about light aimed outward (i.e. trying, but failing, to go outward) rather than light aimed inward.